Calculus PYQ — Page 19

The integral $16{\int }_{1}^{2}\frac{dx}{{x}^{3}{({x}^{2}+2)}^{2}}$ is equal to

The value of $\frac{8}{\pi }{\int }_{0}^{\frac{\pi }{2}}\frac{{(\mathrm{cos}x)}^{2023}}{{(\mathrm{sin}x)}^{2023}+{(\mathrm{cos}x)}^{2023}}dx$ is ______.

Let the function $f(x)=2{x}^{3}+(2p-7){x}^{2}+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then, the set of all values of $p$ is

The area of the region enclosed by the curve $y={x}^{3}$ and its tangent at the point $(–1,–1)$ is

The area of the region ${(x,y):{x}^{2}\leq y\leq 8-{x}^{2},y\leq 7}$ is

Let the area of the region ${(x,y):|2x-1|\leq y\leq |{x}^{2}-x|,0\leq x\leq 1}$ be $A$. Then ${(6A+11)}^{2}$ is equal to _____ .

Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation ${x}^{2}-px+\frac{5}{4}p=0$ are rational. Then the area of the region ${(x,y):0\leq y\leq (x-q{)}^{2},0\leq x\leq q}$ is

The area of the region $A={(x,y):|\mathrm{cos}x-\mathrm{sin}x|\leq y\leq \mathrm{sin}x,0\leq x\leq \frac{\pi }{2}}$

The area enclosed between the curves ${y}^{2}+4x=4$ and $y-2x=2$ is

The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\frac{{x}^{2}+{y}^{2}}{2xy},x>0$. If $y(2)=0$, then a value of $y(8)$ is

Let $y={y}_{1}(x)$ and $y={y}_{2}(x)$ be the solution curves the differential equation $\frac{dy}{dx}=y+7$ with initial conditions ${y}_{1}(0)=0$ and ${y}_{2}(0)=1$ respectively. Then the curves $y={y}_{1}(x)$ and $y={y}_{2}(x)$ intersect at

Let $f$ be a differentiable function such that ${x}^{2}f(x)-x=4{\int }_{0}^{x}tf(t)dt,f(1)=\frac{2}{3}$. Then $18f(3)$ is equal to

If the solution curve $f(x,y)=0$ of the differential equation $(1+{\mathrm{log}}_{e}x)\frac{dx}{dy}-x{\mathrm{log}}_{e}x={e}^{y},x>0,$ passes through the points $(1,0)$ and $(a,2)$, then ${a}^{a}$ is equal to

If the solution curve of the differential equation $(y-2{\mathrm{log}}_{e}x)dx+(x{\mathrm{log}}_{e}{x}^{2})dy=0,x>1$ passes through the points $(e,\frac{4}{3})$ and $({e}^{4},\alpha )$, then $\alpha$ is equal to $_______$

If $y=y(x)$ is the solution curve of the differential equation $\frac{dy}{dx}+y\mathrm{tan}x=x\mathrm{sec}x,0\leq x\leq \frac{\pi }{3},y(0)=1$, then $y(\frac{\pi }{6})$ is equal to

Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma {e}^{-\beta t}$ Where, $\alpha >0,\beta >0$ and $\gamma >0$. Then ${Lim}_{t\rightarrow \infty }y(t)$

The value of the integral ${\int }_{1/2}^{2}\frac{{\mathrm{tan}}^{-1}x}{x}dx$ is equal to

Let $\alpha >0$. If ${\int }_{0}^{\alpha }\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$ then $\alpha$ is equal to :

$\underset{x\rightarrow 0}{\mathrm{lim}}((\frac{1-{\mathrm{cos}}^{2}(3x)}{{\mathrm{cos}}^{3}(4x)})(\frac{{\mathrm{sin}}^{3}(4x)}{{({\mathrm{log}}_{e}(2x+1))}^{5}}))$ is equal to

Let the solution curve $y=y(x)$ of the differential equation $\frac{dy}{dx}-\frac{3{x}^{5}{\mathrm{tan}}^{-1}({x}^{3})}{{(1+{x}^{6})}^{\frac{3}{2}}}y=2x$ $\mathrm{exp}\frac{{x}^{3}-{\mathrm{tan}}^{-1}{x}^{3}}{\sqrt{(1+x{)}^{6}}}$ pass through the origin. Then $y(1)$ is equal to:

${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4{x}^{2}}}dx$ is equal to

If the area of the region ${(x,y):|{x}^{2}-2|\leq y\leq x}$ is $A$, then $6A+16\sqrt{2}$ is equal to ______________

If ${\int }_{\frac{1}{3}}^{3}|{\mathrm{log}}_{e}x|\mathrm{dx}=\frac{m}{n}{\mathrm{log}}_{e}(\frac{{n}^{2}}{e})$, where $m$ and $n$ are coprime natural numbers, then ${m}^{2}+{n}^{2}-5$ is equal to _____ .

The value of $\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{1+2-3+4+5-6+\ldots +(3n-2)+(3n-1)-3n}{\sqrt{2{n}^{4}+4n+3-}\sqrt{{n}^{4}+5n+4}}$ is